As is known, in recent years a clear tendency to shift radio communications systems from single-carrier (SC) to multi-carrier (MC) transmission can be observed. There are several reasons for this paradigm shift, among which the following are worthy of mention:                the necessity to cover wider bandwidths, which is related to augmented data-rate and improved robustness to multi-path fading, can be dealt with more simply by adopting MC transmission. Especially signal equalization can be performed in a largely more effective way in the frequency domain, subcarrier by subcarrier, than in the time domain, when a limited computation power is available;        in multiple antenna systems it is easier to design a multiple input, multiple output (MIMO) receiver with good performance when working on transmission channels that are substantially frequency-flat. This can be achieved by properly choosing the subcarrier spacing in a MC transmission while it cannot be achieved in a wideband SC transmission;        maximum spectrum efficiency implies a process of water-filling to determine bit and power allocation at the transmitter. Bit and power allocation are frequency variant quantities and easy to allocate in a MC transmission system. A water-filling process is instead hardly applicable to SC transmission systems, especially for what concerns bit allocation.        
Given the availability of direct and inverse Fast Fourier Transform (FFT) processors, the majority of the MC transmission systems which are being developed for the communication mass market are based on Orthogonal Frequency Division Multiplexing (OFDM).
OFDM gives an easy way to equalize multi-path fading channels having a frequency band that can exceed 1 GHz in some systems (e.g. IEEE 802.15.3c) and makes MIMO transmission practical in cases where e.g. SC CDMA transmission would make it largely inapplicable. All of these advantages come to the expense of one IFFT and one FFT computation per data block, and a slight drop in frequency efficiency due to the insertion of a cyclic prefix (CP) before each symbol or a zero padding (ZP) after each symbol, which result in differently structured communication signals. For a detailed discussion of the CP and the ZP, reference may for example be made to Bertrand Muquet, Member, IEEE, Zhengdao Wang, Student Member, IEEE, Georgios B. Giannakis, Fellow, IEEE, Marc de Courville, Member, IEEE, and Pierre Duhamel, Fellow, IEEE, “Cyclic Prefixing or Zero Padding for Wireless Multicarrier Transmissions?” IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 12, DECEMBER 2002.
In general, CP and ZP communication modes are different techniques to ensure the same purpose, that is to introduce cyclicity in the received signal (after proper processing in the case of ZP) and to limit inter-block interference. Different communication modes might be used with the same above mentioned purpose, however remaining within the scope of the present invention.
For a single input, single output (SISO) OFDM system with N subcarriers, if the transmitted data after scrambling, channel coding, interleaving and mapping to a constellation is referenced by X (modulation symbol associated with a constellation point in a constellation diagram), the signal at the output of the IFFT may be mathematically expressed as follows:
                                          x            IFFT                    ⁡                      (            t            )                          =                              1            N                    ⁢                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                          X                ⁡                                  (                  n                  )                                            ⁢                                                ⅇ                                      j2π                    ⁢                                                                                  ⁢                                          nt                      /                      N                                                                      .                                                                        (        1        )            
The set of N samples xIFFT(t), t/TS=0 . . . N−1, where TS is the sampling period, constitutes the data part of one OFDM symbol. The final OFDM signal that after amplification is supplied to the transmitting antenna is a sequence of OFDM symbols, where a CP or ZP is inserted between different symbols.
When a CP with length G is used, transmitted data has the following form:xCP(0),xCP(1),xCP(2), . . .   (2)where:xCP(Φ)=[xIFFT(N−G+k),xIFFT(N−G+1+k), . . .. . . ,xIFFT(N−1+k),xIFFT(k),xIFFT(1+k), . . . ,xIFFT(N−1+k)], k=ΦN  (3)and Φ is zero or a positive integer.
When a ZP of length G is used, transmitted data has the following form:xZP(0),xZP(1),xZP(2), . . .   (4)where:xZP(Φ)=[xIFFT(k),xIFFT(1+k), . . . ,xIFFT(N−1+k),01,02, . . . ,0G], k=ΦN  (5)
If the overall transmitted signal is referenced in both cases as x(t), then the received signal takes the following form:
                              r          ⁡                      (            t            )                          =                                            ∑                              l                =                0                                            Δ                -                1                                      ⁢                                                            c                  l                                ⁡                                  (                  t                  )                                            ⁢                              x                ⁡                                  (                                      t                    -                    l                                    )                                                              +                      n            ⁡                          (              t              )                                                          (        6        )            where Δ represents the delay spread of the channel, cl is the complex channel coefficient for the l-th path, and n is the noise.
CP or ZP are necessary to avoid interference in time between different OFDM symbols (which is effective when Δ<G), and to ensure that the data sent to the FFT at the receiver are circularly shifted. This in turn is necessary because even small errors in symbol timing recovery can move the FFT window from its ideal position. However, if data is cyclically shifted, timing errors will simply result in a phase shift after FFT, which can typically be included and corrected in channel estimation and compensation.
When a ZP is used, to ensure cyclic shift it is necessary to perform the following block-wise operation at the receiver:{tilde over (r)}ZP(t)={tilde over (r)}(0),{tilde over (r)}(1),{tilde over (r)}(2), . . .   (7){tilde over (r)}(Φ)=[r(k)+r(N+k),r(1+k)+r(N+1+k), . . . ,r(G−1+k)+r(N+G−1+k),r(G+k), . . . ,r(N−1−k),r(N+k),r(N+1+k), . . . ,r(N+G−1+k)], k=ΦN 
When a CP is used, no operation is required and the received data can be directly sent to the FFT once that time synchronization point is decided:rCP(t)≡r(t)  (8).
CP and ZP methods are basically equivalent as for transmission performance, as both use the same amount of time (channel occupancy overhead) to transmit redundant information.
Advantages of the CP method are:                it makes it available at the receiver a signal with high auto-correlation. Can be used for simple synchronization processing but is essentially useless when the system frame structure includes synchonization-specific information blocks (training and synchronization sequences); and        in certain cases, it might diminish the requirements on signal windowing to limit spectral re-growth at the transmitter.        
Disadvantages of the CP method are:                the average radiated power and possible battery consumption are higher than in the ZP method. However, instantaneous maximum radiated power is the same as in the ZP method. Also the requirements on amplifier range and linearity are the same.        
Advantage of the ZP method is:                it avoids transmission of useless signal section, such saving battery power and lowering average radiated power.        
Disadvantages of the ZP method are:                it is not applicable if the receiver expects a received signal with auto-correlation peaks in coincidence of the CP (which might be the case for some OFDM systems); and        it needs processing at the receiver, although very simple, before FFT.        
In general, the use of many different types of MC transmission, including OFDM, gives way to shortcomings concerning signal amplification. In systems where each subcarrier occupies a different frequency band, if each one has its own power amplifier (PA), the various signals can be seen as SC signals and as such the signal crest factor can be higher than 0 dB only due to the crest factor of the constellation, which is usually increased because of oversampling. The crest factor in such cases is typically limited below a few decibels, and signal power amplification is not particularly critical. On the contrary, in transmitting systems like OFDM, subcarriers overlap in frequency and the overall signal envelope is amplified by a single PA. Unluckily, as different subcarriers carry uncorrelated signals, there is always a non-zero probability that different subcarriers sum up coherently in a certain time instant. Overall, the OFDM envelope can vary noticeably in time. In the following, OFDM is used as a typical example of MC transmission, but the considerations provided apply also to different types of MC transmission.
The above issue is usually measured in terms of the squared crest factor, which is called Peak-to-Average Power Ratio (PAPR). A vast literature is available describing PAPR from OFDM, e.g. H. Ochiai, H. Imai “On the Distribution of the Peak-to-Average Power Ratio in OFDM Signals”, IEEE 2001. In general, PAPR increases for increasing N, but PAPR is a complicated function of N. PAPR is additionally increased by constellation PAPR and oversampling, although the different PAPRs luckily do not sum up linearly, but less than linearly.
A high PAPR means the presence of instantaneous peaks in the signal envelope, which have an amplitude that can be tens of decibels above the average signal power. Real world PA devices have a limited range of input signals they can amplify with high linearity. Above a certain input amplitude, the PA is driven outside its linear region, causing distortion.
If no action is taken to lower PAPR, an OFDM signal, especially when having hundreds of subcarriers or more, and when having oversampling, will have a random distribution in time of peaks with different amplitudes, sometimes very high.
To try and amplify correctly the entire envelope of an OFDM signal, the designer is free to adopt a PA with a linear region much wider than the average signal level of OFDM. In other words, the designer can choose a PA with high input backoff. However, this means higher hardware cost, and is usually not acceptable for consumer products.
For this reason, it is better to try to reduce PAPR in the OFDM signal itself, so that a less expensive PA can be used. Historically, several types of solutions to overcome the PAPR issue in OFDM have been considered, most of which belong to the following categories:
a) clipping: in this case the signal is hard-limited below a certain threshold. Depending on the value of the threshold and on the PAPR value of the original signal, this process brings a certain degree of degradation in the link performance (typically an error floor). Moreover, clipping can cause signal re-growth outside the spectral mask, which is often unacceptable. A description of this technique may for example be found in US 2008/0101502, wherein an optimized clipping for peak-to-average power ratio reduction is disclosed. An optimized clipping pulse is generated which meets certain requirements, such as a spectral mask target or an error vector magnitude (EVM) target (expected standard deviation between signal constellation points before and after PAPR reduction), when applied to a signal;
b) using a certain fraction of the total subcarriers not to carry information data but to reduce the PAPR. This kind of solution inherently reduces spectral efficiency; in fact, it has been reported in the literature that the fraction of unusable subcarriers needed to appreciably lower PAPR is not negligible, e.g. 5 to 10% or more. Moreover, to guarantee a substantial reduction of PAPR, complicated iterative algorithms are necessary at the transmitter, having a computational load and/or delay that is unacceptable in most practical applications. A description of this technique may for example be found in Kamran Haider, “Peak to Average Ratio Reduction in Wireless OFDM Communication Systems”, Degree of Master of Science in Electrical Engineering, Blekinge Institute of Technology, Department of Telecommunications and signal processing, January 2006;
c) phase rotation at the transmitter. In this case the phase of the subcarriers is rotated before IFFT according to a set of pre-determined sequences. IFFT processing is applied to all of the candidate OFDM symbols, and the one with lowest PAPR value is transmitted. The receiver tries to decode the possible OFDM symbols corresponding to all the sequences in a blind way, and can verify the correctness of a certain sequence when channel decoding of CRC-check is error free (blind detection). This technique produces a sustainable increase in the complexity of the transmitter due to multiple IFFTs that are to be performed in parallel, in contrast to the technique previously described in item b), but in general multiplies the complexity of the receiver by the number of sequences, which is not acceptable. Some variants of this technique try to signal what sequence has been used, but this reduces the number of subcarriers available for payload data. A description of this technique may for example be found either in the above-referenced “Peak to Average Ratio Reduction in Wireless OFDM Communication Systems” or in WO 2004/054193, wherein a scrambling-based peak-to-average power ratio reduction without side information is disclosed. One of a set of scrambling sequences is used at the transmitter which results in either a minimum or acceptable peak power. Rather than transmitting side information identifying which of the set of scrambling sequences was selected, the scrambling sequence is also applied to the cyclic redundancy check (CRC), and using syndrome detection the receiver is capable of determining which scrambling sequence was used at the transmitter and can proceed with the necessary de-scrambling. This solution, however, can be used only with a specified type of channel coding, that is in general not acceptable;
d) translation of constellation signals inside their decision region. According to this technique, the value of the constellation signals can be varied iteratively, provided that signals do not go outside their decision region, until the IFFT output converges to a solution having limited PAPR. The technique can be applied in particular to the external signals in constellations, bringing them towards an even more external position. However, this technique involves complex iterative computations as well as multiple IFFTs at the transmitter, and in general does not guarantee an appreciable reduction of PAPR. A description of this technique may for example be found in Cristina Ciochina, Fabien Buda and Hikmet Sari, “An Analysis of OFDM Peak Power Reduction Techniques for WiMAX Systems”, IEEE International Conference on Communications, June 2006, vol. 10, pages 4676-4681.